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Power Analysis for Student's and Welch's T-Tests

means.student.welch.Rd

Calculates power or sample size (only one can be NULL at a time) for Student's and Welch's T-Tests. Welch's T-Test implementation relies on formulas proposed by Bulus (2024).

Use means.to.d() to convert raw means and standard deviations to Cohen's d, and d.to.cles() to convert Cohen's d to the probability of superiority. Note that this interpretation is appropriate only when the underlying distribution is approximately normal and the two groups have similar population variances.

In contrast to previous versions, users can now specify whether their claims will be based on raw score mean difference with P-values or standardized mean difference with confidence intervals. While results typically differ by only a few units, these distinctions can be particularly consequential in studies with small sample sizes or high-risk interventions.

Formulas are validated using Monte Carlo simulations (see Bulus, 2024), G*Power, http://powerandsamplesize.com/, and tables in PASS documentation. One key difference between PASS and pwrss lies in how they handle non-inferiority and superiority tests-that is, one-sided tests defined by a negligible effect margin (implemented as of this version). PASS shifts the test statistic so that the null hypothesis assumes a zero effect, treating the negligible margin as part of the alternative hypothesis. As a result, the test statistic is evaluated against a central distribution. In contrast, pwrss treats the negligible effect as the true null value, and the test statistic is evaluated under a non-central distribution. This leads to slight differences up to third decimal place. To get the same results, reflect the margin in null.d and specify margin = 0.

Equivalence tests are implemented in line with Bulus and Polat (2023), Chow et al. (2018) and Lakens (2017).

NOTE: The functions pwrss.z.mean() and pwrss.z.2means() are no longer supported. The pwrss.t.mean() and pwrss.t.2means() functions are deprecated, but they will remain available as wrappers for power.t.student() or power.t.welch() during the transition period.

Usage

power.t.student(d, null.d = 0, margin = 0,
                n2 = NULL, n.ratio = 1, power = NULL, alpha = 0.05,
                alternative = c("two.sided", "one.sided", "two.one.sided"),
                design = c("independent", "paired", "one.sample"),
                claim.basis = c("md.pval", "smd.ci"),
                ceiling = TRUE, verbose = TRUE, pretty = FALSE)

power.t.welch(d, null.d = 0, margin = 0,
              var.ratio = 1, n.ratio = 1, n2 = NULL,
              power = NULL, alpha = 0.05,
              alternative = c("two.sided", "one.sided", "two.one.sided"),
              claim.basis = c("md.pval", "smd.ci"),
              ceiling = TRUE, verbose = TRUE, pretty = FALSE)

Arguments

d

Cohen's d or Hedges' g.

null.d

Cohen's d or Hedges' g under null, typically 0(zero).

margin

margin - ignorable d - null.d difference.

var.ratio

variance ratio in the form of sd1^2 / sd2^2.

n2

integer; sample size in the second group (or for the single group in paired samples or one-sample).

n.ratio

n1/n2 ratio (applies to independent samples only)

power

statistical power, defined as the probability of correctly rejecting a false null hypothesis, denoted as \(1 - \beta\).

alpha

type 1 error rate, defined as the probability of incorrectly rejecting a true null hypothesis, denoted as \(\alpha\).

alternative

character; the direction or type of the hypothesis test: "one.sided", "two.sided", "two.one.sided". For non-inferiority or superiority tests, add or subtract the margin from the null hypothesis value and use alternative = "one.sided".

design

character; "independent", "paired" or "one.sample".

claim.basis

character; "md.pval" when claims are based on raw mean differences and p-values, "smd.ci" when claims are based on standardized mean differences and confidence intervals.

ceiling

logical; whether sample size should be rounded up. TRUE by default.

verbose

logical; whether the output should be printed on the console. TRUE by default.

pretty

logical; whether the output should show Unicode characters (if encoding allows for it). FALSE by default.

Value

parms

list of parameters used in calculation.

test

type of the statistical test (T-Test).

df

degrees of freedom.

ncp

non-centrality parameter for the alternative.

null.ncp

non-centrality parameter for the null.

t.alpha

critical value(s).

power

statistical power \((1-\beta)\).

n

sample size (`n` or `c(n1, n2)` depending on the design.

References

Bulus, M. (2024). Robust standard errors and confidence intervals for standardized mean difference [Preprint].doi:10.31219/osf.io/k6mbs

Bulus, M., & Polat, C. (2023). pwrss R paketi ile istatistiksel guc analizi [Statistical power analysis with pwrss R package]. Ahi Evran Universitesi Kirsehir Egitim Fakultesi Dergisi, 24(3), 2207-2328. doi:10.29299/kefad.1209913

Chow, S. C., Shao, J., Wang, H., & Lokhnygina, Y. (2018). Sample size calculations in clinical research (3rd ed.). Taylor & Francis/CRC.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.

Lakens, D. (2017). Equivalence tests: A practical primer for t tests, correlations, and meta-analyses. Social psychological and personality science, 8(4), 355-362. doi:10.1177/1948550617697177

Examples

#######################
# Independent Samples #
#######################

## difference between group 1 and group 2 is not equal to zero
## targeting minimal difference of Cohen'd = 0.20
## non-parametric
power.np.wilcoxon(d = 0.20,
                  power = 0.80,
                  alternative = "two.sided",
                  design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Rank-Sum Test (Independent Samples) 
#> (Wilcoxon-Mann-Whitney or Mann-Whitney U Test)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d = 0 
#>   H1 (Alt. Claim) : d - null.d != 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 412 and 412  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## parametric
power.t.student(d = 0.20,
                power = 0.80,
                alternative = "two.sided",
                design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d = 0 
#>   H1 (Alt. Claim) : d - null.d != 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 394 and 394  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 

## when sample size ratio and group variances differ
power.t.welch(d = 0.20,
              n.ratio = 2,
              var.ratio = 2,
              power = 0.80,
              alternative = "two.sided")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Welch's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d = 0 
#>   H1 (Alt. Claim) : d - null.d != 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 473 and 237  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 


## difference between group 1 and group 2 is greater than zero
## targeting minimal difference of Cohen'd = 0.20
## non-parametric
power.np.wilcoxon(d = 0.20,
                  power = 0.80,
                  alternative = "one.sided",
                  design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Rank-Sum Test (Independent Samples) 
#> (Wilcoxon-Mann-Whitney or Mann-Whitney U Test)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= 0 
#>   H1 (Alt. Claim) : d - null.d > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 325 and 325  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 

## parametric
power.t.student(d = 0.20,
                power = 0.80,
                alternative = "one.sided",
                design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= 0 
#>   H1 (Alt. Claim) : d - null.d > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 310 and 310  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## when sample size ratio and group variances differ
power.t.welch(d = 0.20,
              n.ratio = 2,
              var.ratio = 2,
              power = 0.80,
              alternative = "one.sided")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Welch's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= 0 
#>   H1 (Alt. Claim) : d - null.d > 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 372 and 186  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 


## mean of group 1 is practically not smaller than mean of group 2
## targeting minimal difference of Cohen'd = 0.20 and can be as small as -0.05
## non-parametric
power.np.wilcoxon(d = 0.20,
                  margin = -0.05,
                  power = 0.80,
                  alternative = "one.sided",
                  design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Rank-Sum Test (Independent Samples) 
#> (Wilcoxon-Mann-Whitney or Mann-Whitney U Test)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 208 and 208  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 

## parametric
power.t.student(d = 0.20,
                margin = -0.05,
                power = 0.80,
                alternative = "one.sided",
                design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 199 and 199  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 

## when sample size ratio and group variances differ
power.t.welch(d = 0.20,
              margin = -0.05,
              n.ratio = 2,
              var.ratio = 2,
              power = 0.80,
              alternative = "one.sided")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Welch's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 238 and 119  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 


## mean of group 1 is practically greater than mean of group 2
## targeting minimal difference of Cohen'd = 0.20 and can be as small as 0.05
## non-parametric
power.np.wilcoxon(d = 0.20,
                  margin = 0.05,
                  power = 0.80,
                  alternative = "one.sided",
                  design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Rank-Sum Test (Independent Samples) 
#> (Wilcoxon-Mann-Whitney or Mann-Whitney U Test)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 578 and 578  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 

## parametric
power.t.student(d = 0.20,
                margin = 0.05,
                power = 0.80,
                alternative = "one.sided",
                design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 552 and 552  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 

## when sample size ratio and group variances differ
power.t.welch(d = 0.20,
              margin = 0.05,
              n.ratio = 2,
              var.ratio = 2,
              power = 0.80,
              alternative = "one.sided")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Welch's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 662 and 331  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 


## mean of group 1 is practically same as mean of group 2
## targeting minimal difference of Cohen'd = 0
## and can be as small as -0.05 or as high as 0.05
## non-parametric
power.np.wilcoxon(d = 0,
                  margin = c(-0.05, 0.05),
                  power = 0.80,
                  alternative = "two.one.sided",
                  design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Rank-Sum Test (Independent Samples) 
#> (Wilcoxon-Mann-Whitney or Mann-Whitney U Test)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= min(margin) or 
#>                     d - null.d >= max(margin) 
#>   H1 (Alt. Claim) : d - null.d > min(margin) and 
#>                     d - null.d < max(margin) 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 7175 and 7175  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## parametric
power.t.student(d = 0,
                margin = c(-0.05, 0.05),
                power = 0.80,
                alternative = "two.one.sided",
                design = "independent")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= min(margin) or 
#>                     d - null.d >= max(margin) 
#>   H1 (Alt. Claim) : d - null.d > min(margin) and 
#>                     d - null.d < max(margin) 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 6852 and 6852  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## when sample size ratio and group variances differ
power.t.welch(d = 0,
              margin = c(-0.05, 0.05),
              n.ratio = 2,
              var.ratio = 2,
              power = 0.80,
              alternative = "two.one.sided")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Welch's T-Test (Independent Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= min(margin) or 
#>                     d - null.d >= max(margin) 
#>   H1 (Alt. Claim) : d - null.d > min(margin) and 
#>                     d - null.d < max(margin) 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 8222 and 4111  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 


##################
# Paired Samples #
##################

## difference between time 1 and time 2 is not equal to zero
## targeting minimal difference of Cohen'd = -0.20
## non-parametric
power.np.wilcoxon(d = -0.20,
                  power = 0.80,
                  alternative = "two.sided",
                  design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Signed-Rank Test (Paired Samples)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d = 0 
#>   H1 (Alt. Claim) : d - null.d != 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 208  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 

## parametric
power.t.student(d = -0.20,
                power = 0.80,
                alternative = "two.sided",
                design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Paired Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d = 0 
#>   H1 (Alt. Claim) : d - null.d != 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 199  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.198
#>   Statistical Power      = 0.802
#> 

## difference between time 1 and time 2 is less than zero
## targeting minimal difference of Cohen'd = -0.20
## non-parametric
power.np.wilcoxon(d = -0.20,
                  power = 0.80,
                  alternative = "one.sided",
                  design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Signed-Rank Test (Paired Samples)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d >= 0 
#>   H1 (Alt. Claim) : d - null.d < 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 164  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.198
#>   Statistical Power      = 0.802
#> 

## parametric
power.t.student(d = -0.20,
                power = 0.80,
                alternative = "one.sided",
                design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Paired Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d >= 0 
#>   H1 (Alt. Claim) : d - null.d < 0 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 156  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## mean of time 1 is practically not greater than mean of time 2
## targeting minimal difference of Cohen'd = -0.20 and can be as small as 0.05
## non-parametric
## non-parametric
power.np.wilcoxon(d = 0.20,
                  margin = 0.05,
                  power = 0.80,
                  alternative = "one.sided",
                  design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Signed-Rank Test (Paired Samples)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 291  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## parametric
power.t.student(d = 0.20,
                margin = 0.05,
                power = 0.80,
                alternative = "one.sided",
                design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Paired Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 278  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## mean of time 1 is practically greater than mean of time 2
## targeting minimal difference of Cohen'd = -0.20 and can be as small as -0.05
## non-parametric
power.np.wilcoxon(d = 0.20,
                  margin = -0.05,
                  power = 0.80,
                  alternative = "one.sided",
                  design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Signed-Rank Test (Paired Samples)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 105  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.198
#>   Statistical Power      = 0.802
#> 

## parametric
power.t.student(d = 0.20,
                margin = -0.05,
                power = 0.80,
                alternative = "one.sided",
                design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Paired Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= margin 
#>   H1 (Alt. Claim) : d - null.d > margin 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 100  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.199
#>   Statistical Power      = 0.801
#> 


## mean of time 1 is practically same as mean of time 2
## targeting minimal difference of Cohen'd = 0
## and can be as small as -0.05 or as high as 0.05
## non-parametric
## non-parametric
power.np.wilcoxon(d = 0,
                  margin = c(-0.05, 0.05),
                  power = 0.80,
                  alternative = "two.one.sided",
                  design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Wilcoxon Signed-Rank Test (Paired Samples)
#> 
#>   Method       : Guenther
#>   Distribution : Normal
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= min(margin) or 
#>                     d - null.d >= max(margin) 
#>   H1 (Alt. Claim) : d - null.d > min(margin) and 
#>                     d - null.d < max(margin) 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 3589  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 

## parametric
power.t.student(d = 0,
                margin = c(-0.05, 0.05),
                power = 0.80,
                alternative = "two.one.sided",
                design = "paired")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Student's T-Test (Paired Samples)
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : d - null.d <= min(margin) or 
#>                     d - null.d >= max(margin) 
#>   H1 (Alt. Claim) : d - null.d > min(margin) and 
#>                     d - null.d < max(margin) 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size            = 3427  <<
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#>